from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(163177, base_ring=CyclotomicField(630))
M = H._module
chi = DirichletCharacter(H, M([420,338]))
chi.galois_orbit()
[g,chi] = znchar(Mod(25,163177))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(163177\) | |
| Conductor: | \(163177\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(315\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{315})$ |
| Fixed field: | Number field defined by a degree 315 polynomial (not computed) |
First 31 of 144 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{163177}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{64}{315}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{37}{315}\right)\) | \(e\left(\frac{302}{315}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{128}{315}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{163}{315}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{163177}(200,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{43}{315}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{79}{315}\right)\) | \(e\left(\frac{134}{315}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{86}{315}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{16}{315}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{163177}(625,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{128}{315}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{74}{315}\right)\) | \(e\left(\frac{289}{315}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{256}{315}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{11}{315}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{163177}(800,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{29}{315}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{2}{315}\right)\) | \(e\left(\frac{127}{315}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{58}{315}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{128}{315}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{163177}(1250,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{121}{315}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{193}{315}\right)\) | \(e\left(\frac{128}{315}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{242}{315}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{67}{315}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{163177}(1495,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{289}{315}\right)\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{172}{315}\right)\) | \(e\left(\frac{212}{315}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{263}{315}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{298}{315}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{163177}(1600,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{22}{315}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{121}{315}\right)\) | \(e\left(\frac{281}{315}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{44}{315}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{184}{315}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{163177}(2573,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{16}{315}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{88}{315}\right)\) | \(e\left(\frac{233}{315}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{32}{315}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{277}{315}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{163177}(3133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{187}{315}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{241}{315}\right)\) | \(e\left(\frac{26}{315}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{59}{315}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{304}{315}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{163177}(3315,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{274}{315}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{247}{315}\right)\) | \(e\left(\frac{92}{315}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{233}{315}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{58}{315}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{163177}(5000,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{107}{315}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{116}{315}\right)\) | \(e\left(\frac{121}{315}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{214}{315}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{179}{315}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{163177}(6400,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{8}{315}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{44}{315}\right)\) | \(e\left(\frac{274}{315}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{16}{315}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{296}{315}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{163177}(8654,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{122}{315}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{41}{315}\right)\) | \(e\left(\frac{241}{315}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{244}{315}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{104}{315}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{163177}(10292,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{2}{315}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{11}{315}\right)\) | \(e\left(\frac{226}{315}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{315}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{74}{315}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{163177}(10469,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{157}{315}\right)\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{76}{315}\right)\) | \(e\left(\frac{101}{315}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{314}{315}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{139}{315}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{163177}(11960,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{268}{315}\right)\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{214}{315}\right)\) | \(e\left(\frac{44}{315}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{221}{315}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{151}{315}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{163177}(12532,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{45}\right)\) | \(e\left(\frac{173}{315}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{164}{315}\right)\) | \(e\left(\frac{19}{315}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{31}{315}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{101}{315}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{163177}(12800,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{1}{315}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{163}{315}\right)\) | \(e\left(\frac{113}{315}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{315}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{37}{315}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{163177}(13218,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{167}{315}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{131}{315}\right)\) | \(e\left(\frac{286}{315}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{19}{315}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{194}{315}\right)\) | \(e\left(\frac{2}{7}\right)\) |
| \(\chi_{163177}(17530,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{191}{315}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{263}{315}\right)\) | \(e\left(\frac{163}{315}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{67}{315}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{137}{315}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{163177}(23391,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{4}{315}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{22}{315}\right)\) | \(e\left(\frac{137}{315}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{315}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{148}{315}\right)\) | \(e\left(\frac{6}{7}\right)\) |
| \(\chi_{163177}(23811,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{82}{315}\right)\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{136}{315}\right)\) | \(e\left(\frac{131}{315}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{164}{315}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{199}{315}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{163177}(23951,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{298}{315}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{64}{315}\right)\) | \(e\left(\frac{284}{315}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{281}{315}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{1}{315}\right)\) | \(e\left(\frac{6}{7}\right)\) |
| \(\chi_{163177}(25064,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{166}{315}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{283}{315}\right)\) | \(e\left(\frac{173}{315}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{17}{315}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{157}{315}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{163177}(25069,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{59}{315}\right)\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{167}{315}\right)\) | \(e\left(\frac{52}{315}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{118}{315}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{293}{315}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{163177}(26520,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{45}\right)\) | \(e\left(\frac{253}{315}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{289}{315}\right)\) | \(e\left(\frac{239}{315}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{191}{315}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{226}{315}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{163177}(27311,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{61}{315}\right)\) | \(e\left(\frac{17}{45}\right)\) | \(e\left(\frac{178}{315}\right)\) | \(e\left(\frac{278}{315}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{122}{315}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{52}{315}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{163177}(27974,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{269}{315}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{62}{315}\right)\) | \(e\left(\frac{157}{315}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{223}{315}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{188}{315}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{163177}(28095,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{229}{315}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{157}{315}\right)\) | \(e\left(\frac{47}{315}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{143}{315}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{283}{315}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{163177}(28431,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{45}\right)\) | \(e\left(\frac{277}{315}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{106}{315}\right)\) | \(e\left(\frac{116}{315}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{239}{315}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{169}{315}\right)\) | \(e\left(\frac{6}{7}\right)\) |
| \(\chi_{163177}(32090,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{278}{315}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{269}{315}\right)\) | \(e\left(\frac{229}{315}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{241}{315}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{206}{315}\right)\) | \(e\left(\frac{4}{7}\right)\) |